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FIG. 111.
The linear acceleration varies from point to point, but the angular acceleration is the same for all the elements. Therefore the discussion of the problem becomes simpler if we replace the linear acceleration by the angular acceleration.    This may be done by taking the moments of the forces about the axis.    Since dm can move only in a direction perpendicular to the line r, the resultant force dF must be perpendicular to r.    Therefore the magnitude of the moment dG} due to dF, is dG=rdF
7   dv
= ram 
= r2dm  j     (v = rai). at
Therefore the resultant torque acting upon the body, or the sum of the moments due to the forces acting upon all the particles of the body, is
Xm r2 dm.
But by equation (V) G = /<u.    Therefore
7= /   r2dm, Jo
which is the definition of the moment of inertia given in Chapter VII.
186. Comparison.  There is a perfect analogy between motion of pure translation and motion of pure rotation. This is clearly brought out in the following lists of the magnitudes involved in the two types of motion: